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3.6. CONCLUSIONS 111
Partial Proof. If s + g + m 2 = 0 but x is a solution to (3.65), then s(x) +
g(x) + m(x) 2 is of the form “non-negative + zero + strictly positive”, so cannot
be zero.
We can think of (s, g, m) as a witness to the emptiness of the set of solutions
to (3.65). Again, failure to find such an (s, g, m) is a proof of the existence of
solutions provided we trust the correctness of Theorem 26 and the correctness
of the implementation.
3.5.7 Further Observations
1. The methodology outlined in figure 3.3, and indeed that of [CMMXY09],
has the pragmatic drawback that the decomposition computed, as well as
solving the given problem (3.54), solves all other problems of the same
form with the same polynomials and the variables in the same order. For
example, a decomposition which allows us to write down a quantifier-free
equivalent of
will also solve
and even
∀x 4 ∃x 3 p(x 1 , x 2 , x 3 , x 4 ) > 0 ∧ q(x 1 , x 2 , x 3 , x 4 ) < 0 (3.66)
∀x 4 ∃x 3 p(x 1 , x 2 , x 3 , x 4 ) > 0 ∧ q(x 1 , x 2 , x 3 , x 4 ) < 0 (3.67)
∃x 4 ∃x 3 ∀x 2 p(x 1 , x 2 , x 3 , x 4 ) < 0 ∧ q(x 1 , x 2 , x 3 , x 4 ) ≥ 0 (3.68)
The process of Partial Cylindrical Algebraic Decomposition [CH91] can
make the lifting process (right hand side ↑ in Figure 3.3) more efficient, but
still doesn’t take full account of the structure of the incoming quantified
formula.
3.6 Conclusions
1. The RootOf construct is inevitable (theorem 8), so should be used, as
described in footnote 3 (page 59). Such a notation can avoid the “too
many solutions” trap — see equations (3.43) and (3.44). We should find
a way of extending it to situations such as equation (3.42).
2. While matrix inversion is a valuable concept, it should generally be avoided
in practice.
3. Real algebraic geometry is not simply “algebraic geometry writ real”: it
has different problems and needs different techniques.